Abstract:Geometric integration is
the numerical integration of a differential equation, while
preserving one or more of its geometric/physical
properties exactly, i.e. to within round-off error.

Many of these geometric
properties are of crucial importance in physical applications:
preservation of energy, momentum, angular
momentum, phase-space volume, symmetries, time-reversal
symmetry, symplectic structure and dissipation are examples.

In this talk we present a
survey of geometric numerical integration methods for differential
equations. We include some very new and exciting results on the exact
preservation of energy for ODEs as well as PDEs. These results
are related to very classical results obtained by Lax and
co-workers in the early eighties.

Our aim is to make the
review of use for both the novice and the more experienced
practitioner interested in the new developments and directions of
the past decade.